Before we dig into details, let us first summarize what we will analyze in this subsection. The expression of the SGF (REF ) is purely in terms of the inner products of spinors, which we will call the angle couplings. We first rewrite the SGF into an expression (REF ) in terms of the link couplings (REF ), then do a stationary analysis at large spins to find the geometrical interpretation of the link couplings which is illustrated in (REF ) and () [1]}, [2]}. Link couplings encode the conformal geometry of a tetrahedron both in their norms and phases. This geometrical interpretation can be translated back to the angle couplings. However, the map from the angle couplings to link couplings is only surjective but not bijective. We choose a “geometrical gauge” (REF ), which has the most local sense, to fix the angle coupling definition in terms of the link couplings. The geometrical interpretation of the angle couplings is not relevant to the later analysis in this subsection but will be important for Subsection REF  where we investigate the geometrical meaning of the edge amplitudes.
We plug the critical points of the link couplings into the SGF (REF ) and write the semi-classical version of the SGF (REF ) which can be expressed as a Laplace transform of (the exponential of) the Regge action of a tetrahedron. We also give a formal expression of the classical correspondence (REF ) of the SGF. finally, we conclude this subsection in the diagram (REF ) with the relations between the \(\lbrace 6j\rbrace \) -symbol and the SGF as well as their classical and semi-classical correspondences.
